Proposition 1 ( [1]}, Theorem 1) 
For a region \(q \subset \mathbb {R}^n\)  and control \(u\in A\) , let \(\Vert h\Vert ^{q,u}_{\infty }\equiv \sup _{x\in q} |h(x,u)|\)  .
Given a control \(u\in A\) , regions \(q,q^{\prime } \subset \mathbb {R}^n\) , dataset \(\mathrm {D}\) , regression \(\hat{f}\) , and
positive real vectors \(\epsilon \in \mathbb {R}^n\)  and \(\eta \in \mathbb {R}^n\) , then
\(\hat{P}&(q,u,q^{\prime }) = \max _{x\in q} T(q^{\prime } \mid x, u) \nonumber \\&\le \mathbf {1}^{ Im(q,u)}_{\overline{q}^{\prime } (\epsilon +\eta )} \prod _{i=1}^n P[\Vert \hat{f}^{(i)}-f^{(i)}\Vert ^{q,u}_\infty \le \epsilon _i]\prod _{i=1}^n P[|w^{(i)}|\le \eta _i] \nonumber \\& \hspace{59.75078pt} + \prod _{i=1}^n(1-P[\Vert \hat{f}^{(i)} - f^{(i)}\Vert ^{q,u}_\infty \le \epsilon _i]),\) 
\(\check{P}&(q,u,q^{\prime }) = \min _{x\in q} T(q^{\prime }\mid x,u) \nonumber \\& \ge \mathbf {1}^{Im(q,u)}_{X\setminus \underline{q}^{\prime }(\epsilon +\eta )} \prod _{i=1}^n P[\Vert \hat{f}^{(i)}-f^{(i)}\Vert ^{q,u}_\infty \le \epsilon _i] \hspace{71.13188pt} \nonumber \\& \hspace{136.57323pt} \prod _{i=1}^n P[|w^{(i)}|\le \eta _i].\) 
